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## Proof

Let with . For sufficiently small t we have

But giving

If is positive definite, its smallest eigenvalue obeys .So we have

Then, is a local minimizer for f.

We see that a sufficient condition for a local minimizer is and (Hessian) is positive definite. These conditions are very important and should guide us in the choice of an optimization strategy.

Quadratic functions form the basis for most of the algorithms in optimization, in particular for the quasi-Newton method detailed in this paper. It is then important to discuss some issues involved with these functions. Now, if we pose a quadratic objective function

we see that we want to solve

We may assume that the Hessian is symmetric because

So, the unique global minimizer is the solution of the system above if (the Hessian) is spd.

Next: A quasi-Newton method for Up: Definitions and Conditions for Previous: Theorem
Stanford Exploration Project
4/27/2000